If you need to run a full-on desktop environment over ssh, google for 'vnc over ssh'. IIRC, there are a couple different implementations.
A better approach would be to simply launch the specific GUI app you need.
$ ssh -Y firstname.lastname@example.org
[email@example.com ~]$ xclock
Which would, of course, fire up xlock (assuming you have X11 forwarding set up properly on the server side). Replace xclock with the GUI application you need to run.